Devices and methods using the hermetic transform

ABSTRACT

Systems and methods are described using a Hermetic Transform, as well as related transforms, for applications such as directional reception and/or transmit of signals using phased-array devices and systems. The systems and methods an include identifying a direction of arrival for a mobile communicating device. The systems and methods also include the use of a noise conditioning matrix.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a divisional of application Ser. No. 13/788,556,filed Mar. 7, 2013, pending; which claims priority under §119(e) to U.S.Provisional Application No. 61/607,743, entitled “Devices and MethodsUsing the Hermetic Transform and Related Linear Transform Approaches,”filed Mar. 7, 2012, the contents of each of which are incorporated byreference herein in its entirety and for all purposes.

BACKGROUND OF THE INVENTION

This disclosure deals with systems and methods using a HermeticTransform, as well as related transforms, for applications such asdirectional reception and/or transmit of signals using phased-arraydevices and systems. The Hermetic Transform (and related transforms) canbe designed using an array manifold, effectively a complex calibrationresponse vectors from the array in question to signal arrivals fromdifferent directions, whether developed from a mathematical model orfrom collected data, arranged in a particular fashion. The transform canbe utilized for receiver and/or transmit beams to provide narrowermain-lobes than classical methods would typically allow. Furtherbackground about the Hermetic Transform can be found in U.S. Pat. No.8,064,408, incorporated herein by reference in its entirety and for allpurposes.

SUMMARY OF THE INVENTION

Systems and methods are described for using the Hermetic Transform forbeamforming and other purposes. These systems and methods includetechniques for beamforming, noise conditioning, and creating poles andzeroes. As indicated in the incorporated patent, there are manyapplications, including cellular communication systems, e.g., forfinding a direction of arrival of one or more mobile units, but otherapplications include identifying jammer signals and any otherbeamforming application.

The systems can include a plurality of N elements for receiving signals,where N is a natural number greater than 1, and analog to digitalcircuitry for processing the received signals to produce digitalsignals. In each case, the elements can be antennas or other receivers,such as for receiving sound. The systems can operate with cellularcommunications base stations or other wireless systems for identifyingmobile units.

In one embodiment, a processor is provided for processing the digitalsignals by multiplying the digital signals by a noise conditioningmatrix, performing a Hermetic Transform on the noise conditional data,and determining one or more angles of arrival of the received signalsbased on the noise conditioned and transformed data, wherein the noiseconditioning matrix is derived from the covariance of the internalnoise. The processor can further perform a power spectral density frombeam time-series, or time average the square of the magnitude of thetransformed data, and/or prior to the noise conditioning, derive acovariance matrix of the digital signals.

In another embodiment, the processor processes signals includingperforming a Hermetic Transform on the data, multiply the result of thetransform by null transform matrices to create nulls in the spatialtransfer function, wherein the number of nulls M is greater than N. Thenumber of nulls M can be greater than 2N, or up to N squared minus one.The signals can be noise conditioned by a noise conditioned matrixderived from the covariance of internal noise prior to beingtransformed. The at least one of the null transform matrices can beraised to a power R.

In another embodiment, a processor is for receiving signalscorresponding to a plurality of beam directions and forming the signalsin a matrix, for one beam direction, setting values to zero to derive acovariance matrix from signal arrivals for from other beam directionsother than the one direction, and deriving a beam vector from a signalvector and the covariance matrix. The processor can perform thefunctions of setting values to zero to derive a covariance matrix fromsignal arrivals for from other beam directions other than the onedirection, and deriving a beam vector from a signal vector and thecovariance matrix, for a plurality of beam directions.

For each of the systems identified above and elsewhere in thespecification, methods for processing can be performed.

Other features and advantages will become apparent from the followingdescription and drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1-4 are block diagrams of processing steps performed according tomethods described herein.

FIGS. 5-8 are plots showing results from systems and methods describedherein.

FIGS. 9-14 are block diagrams and plots showing use in detecting ajammer.

DETAILED DESCRIPTION

A conventional approach utilized for directional reception is based on aspatial matched filter. The spatial matched filter forms a beam in agiven direction, characterized by a set of direction cosines(α,β_(i),γ), by multiplying each array element signal channel {V_(i)} bythe complex conjugate of the expected response from the given “lookdirection”, and summing the result. The discussion below applies as wellto the reception of vector fields (e.g., electromagnetic waves) as toscalar fields (e.g., acoustic pressure wave fields). However, indicesrelated to vectors components and polarization are suppressed in thetreatment, below, i.e., the fields of the received signal(s) are treatedas scalar.

Assume there are M receiving directions (“look directions” or“beam-steering directions”), with a representative case being the m-thbeam:B(t;α _(m),β_(im);γ_(m))=GΣA _(i)*(R _(i);α_(m)β_(im),γ_(m))exp{−j[ψ(R_(i))}V _(i)(t)

The sum is over the set of receiving N receiving elements in order toform a single beam in the desired look direction. The “G” term is a gainterm used to normalize beam amplitude appropriately.

In practice, one method of determining the expected responses from thevarious arrival directions would be to measure the response for eachdirection empirically. This set of responses is termed an “arraymanifold.” However, as an idealized case, it can be assumed that thearray consists of a set of point-receiving elements at the positions{R_(i)} and that the signal arrivals correspond to plane waves from aset of M discrete look directions corresponding to wave-vectors {k_(m)},in which case an expression for the M beam outputs is as follows:B _(m)(t)=Σexp(−jk _(m) ·R _(i))V _(i)(t)The wave-vector for plane wave arrival k_(m) can be represented byk _(m)=(2π/λ)[cos(α_(m)), cos(β_(m)), cos(γ_(m))]

Again, the sum is over the N receiving elements. For the case of Mreceiving beams, with M look-directions, the above expression can bewritten in terms of a linear (matrix) transformation as follows:B=T*Vwhere for each time sample (t) the output of the beam former is a vectorB with components B_(m), the input samples are arrayed as column vectorwith components V_(i), and the transformation matrix T has elements,T _(mi)=exp(−jk _(m) ·R _(i))

For the above case of plane-wave arrivals, one recognizes that thelinear transformation is essentially a Fourier Transformation in thespace/wave-vector domain. In the case of regularly spaced elements, andlinear or planar geometries, the use of Discrete Fourier Transform andFast-Fourier Transform algorithms can be made to accomplish the abovetransformations with an efficient implementation in Digital SignalProcessing (DSP) hardware. The spatial matched filter yields a maximumsignal to noise ratio in the case of spatially uncorrelated ambientnoise. Further, half-wavelength spacing (inter-element spacing of L=λ/2)of the array produces exactly zero correlation between the noise signalsreceived. Two corollaries of this statement are: (1) the matched-filterbeam-former above is known to be optimum in terms of signal to noiseratio under the stated circumstances, and (2) for an array with moreclosely spaced elements, the ambient noise signals received by the arrayelements become spatially correlated and therefore more ‘signal-like’ interms of the processing approach that one might contemplate.

The condition L=λ/2 represents sampling at the spatial Nyquist Rate, theminimum spatial sampling that will avoid spatial aliasing, which wouldproduce unwanted grating lobes, i.e., strong responses in unwanteddirections away from the beam look-direction. It is known that the beamresponse of such an array to a given wave-vector has a beam shape thatroughly conforms to the expression ΔK ΔR˜1 where ΔK corresponds to thebeam ambiguity (main lobe width) in wave-vector space, and ΔRcorresponds to the array dimension. For example, in the case of aone-dimensional, linear array of point elements of length D, therule-of-thumb for beam-width in radians Δθ is given by:Δθ=λ/Dwhere λ is the acoustic wavelength. The spatial filters (beams) that areformed essentially produce a significant response to signal arrivalsonly within a range of Δθ in angle around a chosen look direction. Theactual response of the beam-former in the case of a linear arraycorresponds to the familiar patterns known from diffraction theory, andthe above formula is sometimes referred to as the diffraction limit onresolution. This rule of thumb is a well-known type of uncertaintyprinciple, taken for granted by system designers, often without regardto the underlying assumptions on which the result rests. The rule ofthumb extends to a two-dimensional, rectangular, planar array, withdimensions D_(x) and D_(y), the angular “beam-widths” of the beam-formerresponse, Δθ_(x) and Δθ_(y), correspond to the x and y dimensions incomparison to the signal wave-length:Δθ_(x) =λ/D _(x)Δθ_(y) =λ/D _(y)

As the dimensions of the array are reduced in relation to the wavelengthof the arriving signal the ability of the array (when conventionallybeam-formed) to discriminate the signal in the desired look-directionfrom interfering signals, noise, and reverberations arriving fromdirections other than the desired look-direction, diminishes. As thearray gets smaller, the width of the beam main-lobe can “fatten”considerably. The spatial gain of the array against spatially isotropicnoise, corresponds to how much of the noise field is rejected by thebeam-forming process, in comparison to the 4π steradians of solid angleoccupied by the noise.

The use of beam-forming is analogous to the use of filtering in thefrequency domain to reject noise which is either spread out across abroad band or which is located at a frequency not corresponding to thatof the signal (for example, use of FFT processing to detect narrow-band,sinusoidal signals in the presence of noise). In terms of a “rule ofthumb”, the array area can be divided into cells that arehalf-wavelength in each dimension; if there are N such cells, theisotropic noise gain is on the order of 10 log₁₀(N) in decibels (dB).The smaller the array (in units of wavelength), the smaller the spatialprocessing gain.

As stated previously, half-wavelength spacing between elements providesmaximum gain for a conventional beam-former. Reduction of side-lobelevel response is accomplished by antenna channel weighting prior toapplication of the linear transform (beam-forming) matrix T. Theresulting expression is given by the following:B _(m)(t)=Σexp(−jk _(m) ·R _(i))[W _(i) V _(i)(t)]where the {W_(i)} are a set of individual weight factors for eachantenna channel and the sum is understood to run over the index i. Theweights are normally real and positive, with a maximum value placed atthe array center, and with the weighting function being designed totaper off towards the array edges. For this reason, the weightingfunction is sometimes called a “taper function.” Reduction in side-loberesponse (10s of dBs) can be accomplished by channel weighting at theexpense of a modest increase in the beam-width of the main lobe, and amodest reduction in spatial gain against spatially isotropic noise. Aflip side of this is that the side-lobe reduction offers mitigation ofstrong interference from directions outside of the beam main-lobe.Hermetic Transform Receive Beam-Forming Algorithm

An improved beam-forming approach can be designed for the case ofspatially oversampled arrays (L<<λ/2). In this case, the FourierTransform is replaced with a different type of transform, termed aHermetic Transform. The version described here is termed decomposablebecause it contains two individual linear transformations that arecombined to form the Hermetic Transform. In the presence of internalsystem noise (electrical element noise) there is also a third transformcomponent used to condition the transform against the influence of suchnoise(s).

To derive the form of the Hermetic Transform, we first note that thegeneration equation involving weighting above can be placed in thefollowing form:B=T*W*Vwhere the weight matrix W is diagonal, with elementsW _(ij) =W _(i)δ_(ij)where W_(i) is the weight for the i-th array element channel, and δ_(ij)is the Kronecker delta function,

δ_(ij) = 1, for  i = j = 0, otherwise

We then formally defineH=T*W

The linear transformation H combines the matched filter matrix T and aweight matrix W into one single linear transformation. So far, this is arearrangement of terms. However, while recognizing that the separable Wmatrix term has an effect on controlling sidelobe levels and mainlobewidth, the question arises whether a weight matrix W can be “designed”so as to improve the resolution of the overall beam-forming process.

For simplicity assume a problem with a single arrival angle parameter θ.We assume M look-directions {θ_(m), m=1, 2, . . . M} and expectedcomplex signal arrivals at each array element (collectively, “the arraymanifold”) {S_(i)(θ_(m)), i=1, 2, . . . N}. A signal matrix E is thusdefined to have elements according to the following expression:Σ_(im) =S _(i)(θ_(m))and the matched filter transformation matrix, T is given byT=Σ ^(H)where the superscript “H” indicates the Hermitian Transpose, orcomplex-conjugate transpose of the matrix.

We pose the following equation for the design of “ideal” beams:HΣ=TWΣ=Σ ^(H) WΣ=Iwhere I is the identity matrix. The interpretation of this expression isas follows: it is desired to “design” a weight matrix W (complex and notnecessarily diagonal) such that the beam-forming transformation H (theHermetic Transform) when applied to m-th signal arrival vector(dimension N×1) with components S_(i) (θ_(m)) produces an (M×1) vectoroutput with a value of “1” in the m-th row and zeros elsewhere. If sucha solution could be determined, it would represent an “ideal”beam-forming transformation, with delta-function beams. However, inpractice, the idea is to find a solution that is as close as possible tosatisfying the above equation according to some objective function to beminimized. Such solution has been derived, as follows:W=(ΣΣ^(H))^(#)Σ(I)Σ^(H)(ΣΣ^(H))^(#)

Here the “#” symbol represents the pseudo-inverse of the bracketedquantity (Gelb notation), and “superscript H” indicates the HermitianConjugate of the quantity. The pseudo-inverse is based on the SingularValue Decomposition (SVD) of the target matrix, with any singular valueshaving magnitude less than a pre-set threshold set to zero so that thepseudo-inverse operates only over the subspace corresponding tosignificant singular vectors, i.e. those with significant singular valuemagnitudes. The generalization of the above expression replaces theidentity matrix, with a desired response matrix p:W=(ΣΣ^(H))^(#)Σ(ρ)Σ^(H)(ΣΣ^(H))^(#)

The above solution corresponds to Rao's concept of a Minimum NormQuadratic Estimator (MINQUE). C. R. Rao, “Estimation of Variance andCovariance Quantities in Linear Models”, J. Stat. Assoc., Issue 3, pp.1818-1818 (March 2010) Volume 67:112-115

Beam-Forming Using Other Array Representations

In the formulation above, data is used directly in the construction ofthe Hermetic Transform. The present invention can also be made to workwith other representations, for time snapshots from an N-element arraycan be transformed using a linear transformation which combines elementsto create fewer or greater numbers of signal vector components. Forexample data from an 8-element array arranged around a circle can becombined into omni, sine, and cosine pattern channels, as is common indirection finding applications. Such arrays can be beam-form. A lineararray of four elements can be transformed using an FFT of arbitrary sizewith zero-padding to form signal vectors of arbitrarily large size.

Noise Conditioning of the Hermetic Transform

In the presence of electrical noise, or other non-ambient noise, a noiseconditioning matrix is developed to reduce uncorrelated white-noise gainand improve robustness, as a spatial pre-processor prior totransformation. If we term H₀ the unconditioned Hermetic Transformresult as presented above, and H₁ the noise-conditioned HermeticTransform, the following equations apply:H ₁ =H ₀ N _(F)N _(F)=cov(Σ)*[cov(Σ)+cov(N)]^(#)where N_(F) is a noise-filtering transformation (matrix) selected tomake the noise-corrupted signals as close to the clean, array manifoldsignal as possible, in a minimum-quadratic-norm sense, and cov( )indicates the spatial, element-to-element covariance operation beingperformed on the array manifold (Σ) and on the internal noise (N).Empirically, performance turns out to be robust with respect to errorsin the noise covariance estimates (cov(N)).Wave-Vector Power Spectrum/Direction Finding

A wave-vector power spectrum (Power Spectral Density or PSD) can be usedto determine a direction of arrival (DOA) to signal sources (emitters).The wave-vector PSD is essentially signal arrival power as a function ofDOA (wave-vector). The methods involved are analogous to PSD methodsinvolving DFT/FFT techniques for time series, with the contradistinctionthat Hermetic Transforms are utilized instead. Examples of such timeseries techniques are the Blackman-Tukey and Periodogram PSD approaches.Peaks in the PSD imply arrivals from specific direction. When applied toradio-frequency signals the determination of arrival angle using PSD istermed radio direction finding, or just “direction finding” (RDF/DF).

There are several direct methods for obtaining the wave-vector PSD. Asshown in FIG. 1, one method involves forming beams first by applying theHermetic Transform to element data vectors, and then processing datafrom each beam to obtain the PSD from the various beam time-series.Referring to FIG. 2, another method includes applying beam-formingfirst, and then time averaging the modulus squared from the complex timeseries of each beam. Referring to FIG. 3, another method utilizes aHermetic Transform derived from an array manifold produced from a set ofvectors formed from signal covariance matrices (each M×M for anM-element array) reshaped into column vectors (dimension M²×1), arrangedin columns, with each column corresponding to a given DOA for themanifold. When this transform is applied to the similarly re-shapedcovariance matrix from a single arrival or from set of incoherent(uncorrelated) arrivals, a wave-vector PSD estimate is produced. Peaksin the PSD produced from any of the above methods correspond to DFestimates of directions-of-arrival for signals arriving at the array.

Spatial Filtering and Interference Nulling

The Hermetic Transform has at least a pseudo-inverse (#), so thatfiltering in the wave-vector (beam-space) domain can be accomplishedusing the following steps. FIG. 4 is a block diagram of an exemplaryimplementation that further includes a noise conditioning step.

-   -   (1) Multiply a complex data “snap-shot” (I&Q samples from each        array element) by the Hermetic Transform Matrix H (after an        optional noise conditioning step as described herein). Each        “snap-shot” vector is an N×1 column vector for an N-element        array. The Hermetic Transform has dimension M×N for N—DOAs in        the array manifold. The result has dimension N×1 (column vector)        in ‘beam-space”.    -   (2) Multiply the result by an M×M filtering matrix (A) The        result is an N×M matrix.    -   (3) Multiply the result of this operation by the pseudo-inverse        of the Hermetic Transform Matrix, H^(#) (dimension N×M). The        result is a column vector N×1.

The above matrices can be pre-multiplied to produce a filter transformwith particular desired characteristics according to the followingexpression:F=H ^(#) ΛH

As an example, an elemental filter matrix can be designed to place onenull in a particular direction (the p-th DOA) would use a Λ matrix whichis the modified identity matrix. The identity matrix has all ones on thediagonal and zeroes elsewhere. In order to introduce a null, one of thediagonal elements (element in the p-th row, p-th column) is set to zero.The resulting filter matrix can be raised to a power (Rth power) inorder to control the strength of the null. This matrix offers theanalogous function of placing a “zero” into the spatial transferfunction. Similarly an elemental filter matrix can be constructed toemphasize signals only in one direction (the p-th DOA) by starting witha Λ matrix that is all zeros, except for the element in the p-th row andp-th column, the value of which is set to unity. The resulting filtermatrix (M×M) can be raised to a power (Rth) in order to control thestrength of what amounts to a “pole” (beam) or “zero” (null) in thespatial transfer function. The matrices can also be normalized, forexample by dividing by the trace of the matrix in order to controlnumerical precision problems.

Such elemental matrices can be multiplied/cascaded to create shaping ofthe array response to signal arrivals from particular directions. Thematrix multiplication operations do not necessarily commute, thereforedifferent permutations of filtering orders using the same set of filtermatrices can produce significantly different results. One of theinteresting results obtained using the above approach is that there canbe more nulls placed in the response pattern than the conventional limitof (N−1) independent nulls for an array with N elements. FIG. 5 showsfive nulls in an otherwise omnidirectional pattern at a singlefrequency, which was accomplished with a 4-element array. One could get10-15 nulls with four (4) antennas, and potentially up to N²−1; i.e.,one could get at least 2N, 3N, or 4N nulls, up to N²−1 with N antennas.

Representative Performance Comparison

A representative problem is presented here in order to illustrateadvantages that can accrue from the use of Hermetic Transformbeamforming in the processing of signals arriving at a small array underrealistic conditions.

Consider as an example, a circular underwater acoustic array of diameter0.21 meters (8.3 inches) with 18 elements arranged at uniform intervalsaround the circle. Similar results can be obtained for systems ofantenna elements operating at radio and microwave frequencies.

From the formula λ=c/f, c being the speed of sound waves in the mediumand f being the frequency, one computes that this array has a diameterof approximately 0.14λ, for a speed of sound in sea water of 1524 M/sec(5000 feet/sec) at a frequency of 1000 Hz. A comparison of the beamamplitude pattern resulting from the use of a Hermetic Transform vs. aconventional phased-array beam-forming with no amplitude shading wasobtained using a calculation in a MATLAB™ program.

The comparison showed an oval plot with the Hermetic Transformbeam-response to a 1000 Hz plane wave impinging of the array with adirection of arrival of 0 degrees (seen to correspond to the maximumresponse of the beam) while a conventional phased-array beam-formingapproach provided a substantially circular plot shows the correspondingresponse of beam-patterns produced by the Hermetic Transformbeam-forming approach for this same array over a set of frequenciesspanning 1000-5000 Hz provided multiple oval lobes.

This result indicates that the beam-shape is relatively frequencyinsensitive over this range of frequencies where the array diameter issubstantially less than a wavelength.

Detection performance with respect to simulated signals arriving in thepresence of simulated ambient noise interference, demonstratesimprovement (for Hermetic vs. conventional Beam-Forming) in the metricsof detection probability and probability of false alarm which correspondto the predicted improvement spatial noise gains. The latter arecalculated based on predicted beam patterns for both the HermeticTransform approach and the Conventional phased-array approach. For asimple case of an arriving monochromatic complex sinusoidal place wavearriving at the array described above in the presence of an isotropicnoise field consisting of independent Gaussian noises arrival from alldirections (360 degrees), curves were derived from a MATLAB-basedsimulation.

The curves show a cumulative probability of detection for signalarrivals and noise only arrivals versus detection threshold whenconventional beam-forming is applied to 1000 Hz signal having a −6 dBsignal to noise ratio at each element of the 18-element array. Thecurves tests show the corresponding results for the case of HermeticTransform beam forming of the same array. The increase in detectionperformance resulting from the application of the Hermetic Transformbeam forming process, as observed in the simulation experiment,corresponds to an extra spatial gain of approximately 12 dB in signal tonoise ratio. Curves are shown in the provisional application.

A visible graphic representation of the improvement in detectionperformance for the case described above is shown in the provisionalapplication. Two plots are shown, which represent a plot of energy vs.time at the output of the conventional, phased-array beam formingprocess and the Hermetic Transform beam forming Process. The x-axis isbearing (direction of arrival), the y-axis is time, as an ensemble oftime records are presented, and the energy of each beam (bearing) isrepresented by intensity and pseudo-color. Visual detection of a signalin this representation is accomplished by observing differences in theintensity and color of the data being presented relative to thesurrounding ambient background. The conventional plot shows nodifferentiation between signal and isotropic noise, while the HermeticTransform plot shows a band of energy at a particular bearing,corresponding to the direction of the signal arrival. The signal arrivalis at −6 dB (array element) signal-to-noise ratio (SNR). The improvementin visual detection of the signal arrival illustrates a benefit of thegain provided by the Hermetic Transform Beam-Forming of the array ascompared to the conventional phased array, which offers little benefitfor an array this small in comparison to the wavelength of the signal.

Effect of the Noise Conditioning Matrix on the Hermetic Beam-Forming

The discussion above justifies a need for dealing with the case ofinternal system noise that is spatially uncorrelated, element toelement, using a noise filtering or noise conditioning matrix. Theeffect of this noise conditioning matrix has been measured using realdata and shown to offer value in producing a robust Hermetic Transformin the presence of such internal noise. FIG. 6 indicates a rectilinearrepresentation of the beam-pattern of a 7-element array having a singleplane-wave arrival which is observed in the presence of significantinternal noise (˜−10 dB signal to internal noise ratio). The arraylength is 0.67 lambda. The ideal pattern predicted based on use of theunconditioned Hermetic transform on a noise-free signal (DOA=0 degrees)is shown in figure with one large central peak. The measured patternobserved using the unconditioned Hermetic Transform in the presence ofthe internal noise is also shown with five peaks. Finally, the measuredpattern observed when the Hermetic Transform is conditioned using theprocedure described above is applied, is shown in the figure with “dots”representing measured data. The noise conditioning is shown to restorethe pattern to essentially the “ideal” value. The benefits of theHermetic Transform are not altered by adding this additional step ofnoise conditioning.

FIGS. 1-4 are block diagrams of systems described elsewhere herein, andeach includes a noise conditioning step before the Hermetic Transform.

Related Linear Transformations (WRING Transform)

An alternative formulation of a linear transformation that producessimilar results to that of the Hermetic Transform, at least to withinfinite precision of the calculations, has been discovered. Thistransform has been termed the “WRING Transform.”

One formulation of an optimal spatial matched filter for beamforming, isknown in the adaptive beam forming literature as the Direct MatrixInverse (DMI) method. In this method, an estimate of the inverse of thespatial covariance matrix of the noise interference, such interferenceconsisting of a set of background signals and noise which one desires toreject through beam forming, is multiplied by the input signal vectorbefore applying phases shifts and summing to form the beam. According toMonzingo and Miller, “Introduction to Adaptive Arrays”, Section 3.3.2, abeam β(θ_(m)) in the direction of θ_(m) can be formed using DMI usingthe signal reference vectors V(θ_(m)) as previously described and theinverse of the interference covariance matrix R_(nn) ⁻¹ according to theexpressionβ(θ_(m))=[V(θ_(m))]^(H) R _(nn) ⁻¹

The first term on the right-hand-side (RHS) of the above equation is thespatial matched filter to the arriving signal; the second term is aspatial “pre-whitener” of the interfering background.

As described in Monzingo and Miller, the interference covariance matrixis formed from the time averaged outer product of the HermitianConjugate (complex conjugate transpose) of the received data vector withthe received data vector for the case where noise/interference only ispresent.

If the signal and noise interference are not simultaneously present, sothat the interference statistics (covariance matrix) can be estimatedwithout signal contamination (for example in the case of time-slotted orpulsed signals) and the noise interference is wide-sense stationary,then the DMI technique can be effective.

The present method applies similar mathematical reasoning to that usedto derive the DMI to develop a different approach, a beam-formingtransform which is fixed for a given array configuration and signalfrequency, and which is not dependent on the specific noise interferenceand thus need not be data-adaptive.

The essence of the transform generation is as follows:

For each beam index {j}, for j=1, 2, . . . , (Number of Beams)Σ(θ₁,θ₂,θ₃, . . . )=∥V(θ₁)V(θ₂)Vθ ₃) . . . ∥^(H)

-   -   (1) form a Matrix    -   (2) Set all values in row {j} to be zeros, set this matrix=σ    -   (3) Form a covariance matrix,        R=σ ^(H)σ    -   (4) The j-th row of the Transform Matrix T is given by        [V(θ_(j))]^(H) R ^(#)

The concept of this approach is that signals that would correspond toall other beams being formed are treated as sources of interference(noise) in the creation of a noise/interference covariance matrix; thematrix inverse in the conventional DMI technique is replaced with thepseudo-inverse (# symbol in the above equation).

-   -   (5) The beam vector β(θ_(m)) are formed from a signal vector u        using the beam-forming transform matrix multiplied by the signal        vector, according to the equation        ρ(θ_(m))=Tu

The success of the above technique is based on a discovery that thetransformation matrix which results is in most cases nearly identical tothat which results from the Hermetic Transform approach, which, in turnis designed to achieve maximum spatial gain. FIG. 7 provides an exampleproduced using a MATLAB™ analysis/simulation. The simulated array is aone-dimensional, linear array with eleven (11) receiving elements, withthe array being one wavelength (lambda) in length. The elements areideal point elements with no appreciable aperture. The curve shown withone broad main peak is the amplitude pattern of the broadside beamgenerated with conventional Discrete-Fourier Transform processing. Thepattern with a sharper main peak was generated using the DiscreteHermetic Transform. The points correspond to the pattern generated usingthe technique of the present invention, the MSG beam.

Beamforming of Spatially Interpolated Arrays

Both the Discrete Hermetic Transform (DHT) and MSG Transform (MGST) forbeamforming or spectrum analysis, require the signal to be over-sampled,with elements less than half wavelength spacing in the case of beamforming, or with sampling rates faster than the Nyquist rate, in thecase of time or frequency domain processing.

In the case of beamforming of antenna arrays, there is a real-worldconstraint on the number of physical antenna elements and/or theassociated analog-to-digital converter channels may be limited to arelatively small number. These constraints limit the achievable gain inresolution derivable from higher resolution methods such as DHT or MSGTwhich perform better (achieve higher resolution and array gain) with ahigher degree of spatial oversampling. Therefore a useful addition toHermetic Transform and/or MSG beamforming approach is a type ofinterpolation used to create signals synthetically which “fill-in”missing data using a smaller number of inputs from physical elements.This discussion is readily extended to spectrum analysis.

A typical result involving the increase of resolution from addingadditional physical elements for an array of a fixed physical dimension(aperture size) is shown in FIG. 8. A calculation of the conventionalamplitude (e.g. voltage response, not power) beam-pattern of an array ofone wavelength (lambda=1), is shown in the plot immediately below (blackcurve). With the conventional resolution formula, θ=λ/D, a linearlydisposed array which is one-lambda in length (θ=λ/D=1) should produce abeam of extent approximately one radian, (about 57 degrees). Byinspection, it is seen that the black curve in the plot indicates a beamwhich is approximately this extent, the other curves shown in the figurebelow indicate amplitude beam patterns calculated with 2-degreeresolution using Hermetic Transform Beam Forming on linear arrays of thesame aperture (D=one-lambda) for various numbers of physical elements.The minimum number of elements needed to satisfy the spatial Nyquistcriterion is 3 (half-wavelength spacing).

In FIG. 8, results are shown for patterns corresponding to conventional,5 elements, 7 elements, and 21 elements, each of the latter three curvescorresponding to different degrees of spatial oversampling, and sharingan increasingly and successively narrow main peak. The 21-elementpattern corresponds to 10-times oversampling relative to the Nyquistsampling criterion for the minimum sampling required to avoid gratinglobes (spatial aliasing). It can be seen from the plot that thebeam-width becomes smaller as the number of elements increases, however,at some point, a minimum beam width is reached. In this case, thebaseline Hermetic Transform Algorithm reaches a minimum width of 14degrees, approximately ¼ of the conventional beam width.

With sampling above Nyquist, intermediate spatial samples can bedetermined, and this array consisting of actual plus synthetic samplescan be used to increase resolution with the Hermetic Transform.

A useful limiting case is one where every calibration vector direction(θ_(k)) provides a separate interpolation matrix, ending up with a setof interpolation matrices, {M_(i), i=1, 2, . . . N_(B)} where N_(B) isthe number of calibration vector directions. A general interpolationmatrix can be formed by taking an appropriate linear combination of theindividual interpolation matrices, each of which maps a P-element outputthat the array would see for a particular direction (θ_(k)) into aQ-element array output for the identical direction (θ_(k)).M=Σ _(i) C _(i) M _(i)

It practical applications there various methods available for generatingthe linear combining coefficients, {C_(i)}: For the case of a P-elementvector signal vector, S, with components S_(j), which represents atime-snapshot of P outputs from a P-element array being interpolated tocreate Q synthetic elements, with Q>P, the i-th interpolation matrixweighting coefficient C is determined according to the wave-numberspectrum content of the original signal in the i direction. One approachis to map the input signal array vector onto the N_(B) directions usingthe Hermetic Transform and add together the interpolated arrays for eachdirection weighted by the component of signal in each direction. Anotherapproach is to pre-filter the signal using a beam transform, then applythe pre-determined spatial filter response as a set of weightingcoefficients.

Simulation results of the process are shown in the provisionalapplication. The results of a 4 to 16 element array synthesis, followedby beamforming with a Hermetic Transform appropriate to a 16-elementarray (with dots) as compared to the ideal case result of Hermeticbeamforming a 16-element array showing the narrowest peak, and Hermeticbeam forming the 4-element array directly with a broader peak for aone-lambda length linear array.

The main lobe of the synthetic beam is effectively preserved, whileside-lobes are maintained at a similar level to that obtained with afull complement of 16 elements.

The process works because, with spatial oversampling with even 4elements in a one-lambda length array, the intermediate spatial samplescan be effectively recovered (synthesized) using interpolation.

Hermetic Beam-Forming for Signal Transmission Sample Results with anN-Element Antenna Array

Prior U.S. Pat. No. 8,064,408 focuses more on receiving. A modifiedversion of the above approach can be applied to transmit arrays.Beginning with a standard expression of the electric field (E) from anarray excited with a specified current density, is as seen below,

one can design a set of current densities for the set of antennaelements in the array in order to create a pattern as narrow aspossible. With respect to the above equation, the following notationsare made:

Reciprocity

-   -   Note: previous result corresponds to standard notation with        j=−i=−(−1)^(1/2)    -   Transmitting Array Excitation is normally the complex conjugate        of array receiving manifold {Σ(θ,φ)} . . . from reciprocity    -   Note: (polarization index suppressed).    -   k=k(θ,φ); assume (θ,φ) are discrete {θ_(m),φ_(n)}    -   Standard Equation is B(θ,φ)=Σ^(H)F    -   B is beam amplitude pattern, F is the Fourier Matrix from the        radiation integral    -   Replace Fourier integral with discrete sum

A general result which uses a transmitting Hermetic Transform has beendeveloped which is stated below:Σ ^(H) W F =βW =(Σ Σ ^(H))^(#) Σ β( F ^(H))( F F ^(H))^(#)

The F is the transmit Greens Function matrix, the Σ^(H) term correspondsto the excitations obtained by reciprocity from the array receivingresponse manifold, i.e., the complex conjugate excitations, and theweight matrix W is used in the transformation of the conjugateexcitations to produce a set of excitations which produce the responseβ. For the maximum directivity, β=c I.

To convert from current to voltage, we need to first obtain the mutualimpedance matrix Z for the array from data or model and form therelationship V=I Z to derive the appropriate voltages for driving thearray. Both the manifold and the mutual impedance can be derived using amodel such as NEC-4 or can be derived empirically.

Representative Transmit Beam Results

The improved transmit directivity which results from this approach isillustrated by the beam pattern shown in the provisional applicationcalculation below. The array is a 24-element circular array of dipoleantennas (or monopoles) over a perfect ground plane with an arraydiameter of one quarter wavelength. The conventional equivalent isominidirectional. One pattern is in decibels (dB) while the otherpattern is power.

Processing Apparatus

In all the embodiments and implementations shown here, processing can bedone with any form of suitable processor, whether implemented inhardware or software, in application-specific circuitry, or specialpurpose computing. Any software programs that are used can beimplemented in tangible media, such as solid state memory or disc-basedmemory. The forms of processing can generally be considered to be“logic,” which can include such hardware or software of combinedhardware-software, implementations. The computing can be done on astandalone processor, groups of processors, or other devices, which canbe coupled to memory (such as solid state or disc-based) for showinginput and output data, and provide output, e.g., via screens orprinters.

Systems of this type are shown in more detail, for example, in U.S. Pat.No. 8,064,408, e.g., at FIGS. 2 and 4.

Creation of Spatial Filters Using Hermetic Transforms

The Hermetic Transform can be used to create elemental spatial filterswhich contain a single direction of signal rejection (nulling)(analogous to a “zero” in the transfer function of a conventionaldigital filter) or a single direction of signal enhancement (analogousto a “pole” in the transfer function of a conventional digital filter).Referring to FIG. 9, the system under study makes us of four (4) antennachannels with coherent sampling of in-phase and quadrature (I&Q)components of the data. Therefore each elemental transform is a complex,4×4 matrix (4-rows, 4 columns). One elemental “pole” and one elemental“zero” were created for each of the look directions in the manifold (forthis study, there are 36 look directions spaced 10 degrees apart,therefore 72 elemental transforms). This process can be accomplishedoffline to create a set of 4×4 matrices, each of which satisfies aparticular specification (rejection or enhancement of signals from onedirection.)

Combining of transforms from the elemental sort into more complexarrangements can be used to create spatial filters according to adesired specification. For simplicity, the specification for the spatialresponse of each filter has “pass-sectors” (analogous to “pass-bands” inconventional digital filtering) each with a desired amplitude of 1, and“reject-sectors” (analogous to “stop-bands” in conventional digitalfiltering). A genetic algorithm is used to create these arrangements,which end up being (still) 4×4 matrices.

FIG. 10 indicates an off-line procedure for making more general spatialtransforms. The current genetic algorithm uses a set of the original 72transform matrices plus all possible sums and products of pairs of thesetransforms. The genetic algorithm guarantees a reasonable solutionwithout the combinatorial explosion associated with a search of allpossible solutions.

Experiment Description

The methodology for the study has been to create simulations which allowcomplete control in MATLAB, supplemented by processing of datacollections taken with the existing 4-channel test bed. The existingtest-bed is a dual band system which operates in at cellularfrequencies. The simulations to date have been conducted for an RFcarrier frequency of 900 MHz. The 4-channel antenna is approximately 3inches in diameter. The figure below indicates the experimental setupfor the general MATLAB simulation.

Referring to FIG. 11, an array model is applied to four-channel datafrom both a given communications signal (“COMM”) and interference/jammersignal (“JAM”) and summed, then added to simulated internal systemnoise. The internal noise is set so that the signal to internal noiseratio is +20 dB. The jammer signal strength is arbitrary. Two testpoints are set up to measure the result of the nulling. A best set ofomni-weights is derived as described above, and applied to the fourchannels at Test Point “in” (prior to applying the nulling matrix), andto the output of the nulling step to create Test Point “out”.

The genetic algorithm supplies a plot of the desired spatial responsewhich can be compared to the desired/specified spatial pattern.Referring to FIG. 12, for a null at zero, a pattern was derived as aprediction of the genetic algorithm, based on calibration from noiseless10-bit LFSR/MSK signals used as the COMM signal (below). The pattern ismeasured in decibels (dB) from the pattern minimum, located at the nullof the pattern. The desired (ideal) pattern is shown with a sharp wedgepattern. The depth of null as shown is approximately 70 dB.

Referring to FIG. 13, the processing performed on the “in” data and“out” data is shown. The jammer data is chosen as 11-bit LFSR/MSK whichoccupies the same spectrum as the signal. The combined signal is matchedfiltered (replica correlated) at both the “in” Test Point and “out” TestPoint, using both the signal replica and the jam signal replica, inorder to determine the signal gain and the jammer gain through theprocess. The outputs of the matched filters (replica correlators) areconverted to power (magnitude squared of the complex data). Thesignal-to-jammer gain is the signal gain minus the jammer gain. Theresults are compiled in dB. The replica correlator is known to be theoptimal processor for the case where the noise is white and Gaussian; ifthis were not the case a noise pre-whitener would be employed to achievemaximum SNR.

Processing Steps to Determine Signal Gain and Jammer Gain

The results obtained from the HCW simulation are shown in FIG. 14.Signal Gain, Jammer Gain, and combined (S/J Gain) are shown. The x-axisis the position of the signal, while the jammer remains fixed at 0degrees, where the null has been placed. A significant gain against thejammer is observed, matching prediction based on the null patternresponse alone. The jammer gain is seen to remain constant (as itshould) as the signal is moved, at approximately −70 dB.

The HPMV simulation verifies the null value, but shows a jammer gainwhich moves with the signal, this being attributed to non-zerocorrelation between the 10-bit LFSR signal for the COMM signal and the11-bit LFSR signal for the JAM. An adjustment to that simulation isbeing made to correct this issue. Another artifact that results fromthis correlation is the appearance of S/J gain varying with jammerpower. Since the process is linear (the null transform is a simple 4×4matrix multiply) the gain should be a constant. The jammer gain showingdependence on signal position and also showing dependence on jammerpower is proof of artifacts in the prior simulation results.

The genetic algorithm produces equally good patterns for nulling in theother directions, for example, the pattern below derived for a nullposition at 270 degrees, giving a −70 dB null. Results with the HCWsimulation thus how that the performance in the other directions is thesame and is only dependent on the relative position of null and jammerand does match what is expected based on the beam patterns of each givennull.

Other embodiments are within the claims. For example, while thedisclosure mainly discusses antennas, other signal receivers could beused, and with data other than I&Q data.

What is claimed is:
 1. A system comprising: a plurality of N receiverelements for receiving signals; analog to digital circuitry forprocessing the received signals to produce digital signals; processorfor processing signals including performing a Hermetic Transform on thedata, multiply the result of the transform by null transform matrices tocreate nulls in a spatial transfer function, wherein a number of nulls Mis greater than N.
 2. The system of claim 1, wherein the elementsinclude antennas.
 3. The system of claim 1, wherein the number of nullsM is greater than 2N.
 4. The system of claim 1, wherein the signals arenoise conditioned by a noise conditioned matrix derived from thecovariance of internal noise prior to being transformed.
 5. The systemof claim 1, wherein at least one of the null transform matrices israised to a power R.
 6. The system of claim 1, wherein the system isoperatively coupled to a cellular base station for identifying adirection of arrival of one or more mobile units.